Powers of Binomials

1. Section 4-2
Powers of Binomials

2. Essential Questions
• How do you use Pascal’s Triangle to expand
powers of binomials?
• How do you use the Binomial Theorem to
expand powers of binomials?

3. Vocabulary
1. Pascal’s Triangle:

4. Vocabulary
1. Pascal’s Triangle: A pattern of numbers that
can be used to determine the coefficients to
expand a binomial (a + b)n

5. Vocabulary
1. Pascal’s Triangle: A pattern of numbers that
can be used to determine the coefficients to
expand a binomial (a + b)n
1

6. Vocabulary
1. Pascal’s Triangle: A pattern of numbers that
can be used to determine the coefficients to
expand a binomial (a + b)n
1
1 1

7. Vocabulary
1. Pascal’s Triangle: A pattern of numbers that
can be used to determine the coefficients to
expand a binomial (a + b)n
1
1 1
1 12

8. Vocabulary
1. Pascal’s Triangle: A pattern of numbers that
can be used to determine the coefficients to
expand a binomial (a + b)n
1
1 1
1 12
3 31 1

9. Vocabulary
1. Pascal’s Triangle: A pattern of numbers that
can be used to determine the coefficients to
expand a binomial (a + b)n
1
1 1
1 12
3 31 1
1 4 4 16

10. Vocabulary
1. Pascal’s Triangle: A pattern of numbers that
can be used to determine the coefficients to
expand a binomial (a + b)n
1
1 1
1 12
3 31 1
1 4 4 16
10 105 51 1

11. Example 1
Expand (p +t)5

12. Example 1
Expand (p +t)5
(p +t)(p +t)(p +t)(p +t)(p +t)

13. Example 1
Expand (p +t)5
(p +t)(p +t)(p +t)(p +t)(p +t)
(p2
+ 2pt +t2
)(p2
+ 2pt +t2
)(p +t )

14. Example 1
Expand (p +t)5
(p +t)(p +t)(p +t)(p +t)(p +t)
(p2
+ 2pt +t2
)(p2
+ 2pt +t2
)(p +t )
(p4
+ 2p3
t + p2
t2
+ 2p3
t + 4p2
t2
+ 2pt 3
+ p2
t2
+ 2pt 3
+t 4
)(p +t)

15. Example 1
Expand (p +t)5
(p +t)(p +t)(p +t)(p +t)(p +t)
(p2
+ 2pt +t2
)(p2
+ 2pt +t2
)(p +t )
(p4
+ 2p3
t + p2
t2
+ 2p3
t + 4p2
t2
+ 2pt 3
+ p2
t2
+ 2pt 3
+t 4
)(p +t)
(p4
+ 4p3
t + 6p2
t2
+ 4pt 3
+t 4
)(p +t )

16. Example 1
Expand (p +t)5
(p +t)(p +t)(p +t)(p +t)(p +t)
(p2
+ 2pt +t2
)(p2
+ 2pt +t2
)(p +t )
(p4
+ 2p3
t + p2
t2
+ 2p3
t + 4p2
t2
+ 2pt 3
+ p2
t2
+ 2pt 3
+t 4
)(p +t)
(p4
+ 4p3
t + 6p2
t2
+ 4pt 3
+t 4
)(p +t )
p5
+ 4p4
t + 6p3
t2
+ 4p2
t 3
+ pt 4
+ p4
t + 4p3
t2
+ 6p2
t 3
+ 4pt 4
+t5

17. Example 1
Expand (p +t)5
(p +t)(p +t)(p +t)(p +t)(p +t)
(p2
+ 2pt +t2
)(p2
+ 2pt +t2
)(p +t )
(p4
+ 2p3
t + p2
t2
+ 2p3
t + 4p2
t2
+ 2pt 3
+ p2
t2
+ 2pt 3
+t 4
)(p +t)
(p4
+ 4p3
t + 6p2
t2
+ 4pt 3
+t 4
)(p +t )
p5
+ 4p4
t + 6p3
t2
+ 4p2
t 3
+ pt 4
+ p4
t + 4p3
t2
+ 6p2
t 3
+ 4pt 4
+t5
p5
+ 5p4
t +10p3
t2
+10p2
t 3
+ 5pt 4
+t5

18. Binomial Theorem
(a + b)n
= nC0an
b0
+ nC1an−1
b1
+ nC2an−2
b2
+...+ nCna0
bn
When expanding any binomial, we can use the pattern

19. Binomial Theorem
(a + b)n
= nC0an
b0
+ nC1an−1
b1
+ nC2an−2
b2
+...+ nCna0
bn
When expanding any binomial, we can use the pattern
a + b( )n
=
n!
k!(n − k )!
an−k
bk
k =0
n
∑

20. Binomial Theorem
(a + b)n
= nC0an
b0
+ nC1an−1
b1
+ nC2an−2
b2
+...+ nCna0
bn
When expanding any binomial, we can use the pattern
This uses Pascal’s Triangle!
a + b( )n
=
n!
k!(n − k )!
an−k
bk
k =0
n
∑

21. Binomial Theorem
(a + b)n
= nC0an
b0
+ nC1an−1
b1
+ nC2an−2
b2
+...+ nCna0
bn
When expanding any binomial, we can use the pattern
This uses Pascal’s Triangle!
The powers of the first part count down.
a + b( )n
=
n!
k!(n − k )!
an−k
bk
k =0
n
∑

22. Binomial Theorem
(a + b)n
= nC0an
b0
+ nC1an−1
b1
+ nC2an−2
b2
+...+ nCna0
bn
When expanding any binomial, we can use the pattern
This uses Pascal’s Triangle!
The powers of the first part count down.
The powers of the second part count up.
a + b( )n
=
n!
k!(n − k )!
an−k
bk
k =0
n
∑

23. Binomial Theorem
(a + b)n
= nC0an
b0
+ nC1an−1
b1
+ nC2an−2
b2
+...+ nCna0
bn
When expanding any binomial, we can use the pattern
This uses Pascal’s Triangle!
The powers of the first part count down.
The powers of the second part count up.
The powers within the term must add up to n.
a + b( )n
=
n!
k!(n − k )!
an−k
bk
k =0
n
∑

24. Example 2
Expand (t −w)8
Binomial Theorem

25. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= 8C0t8
(−w)0

26. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= 8C0t8
(−w)0
+8C1t7
(−w)1

27. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= 8C0t8
(−w)0
+8C1t7
(−w)1
+8C2t6
(−w)2

28. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= 8C0t8
(−w)0
+8C1t7
(−w)1
+8C2t6
(−w)2
+8C3t5
(−w)3

29. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= 8C0t8
(−w)0
+8C1t7
(−w)1
+8C2t6
(−w)2
+8C3t5
(−w)3
+8C4t 4
(−w)4

30. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= 8C0t8
(−w)0
+8C1t7
(−w)1
+8C2t6
(−w)2
+8C3t5
(−w)3
+8C4t 4
(−w)4
+8C5t 3
(−w)5

31. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= 8C0t8
(−w)0
+8C1t7
(−w)1
+8C2t6
(−w)2
+8C3t5
(−w)3
+8C4t 4
(−w)4
+8C5t 3
(−w)5
+8C6t2
(−w)6

32. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= 8C0t8
(−w)0
+8C1t7
(−w)1
+8C2t6
(−w)2
+8C3t5
(−w)3
+8C4t 4
(−w)4
+8C5t 3
(−w)5
+8C6t2
(−w)6
+8C7t1
(−w)7

33. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= 8C0t8
(−w)0
+8C1t7
(−w)1
+8C2t6
(−w)2
+8C3t5
(−w)3
+8C4t 4
(−w)4
+8C5t 3
(−w)5
+8C6t2
(−w)6
+8C7t1
(−w)7
+8C8t0
(−w)8

34. Example 2
Expand (t −w)8
Binomial Theorem

35. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= t8

36. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= t8
−
8!
1!7!
t7
w1

37. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= t8
−
8!
1!7!
t7
w1
+
8!
2!6!
t6
w2

38. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= t8
−
8!
1!7!
t7
w1
+
8!
2!6!
t6
w2
−
8!
3!5!
t5
w 3

39. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= t8
−
8!
1!7!
t7
w1
+
8!
2!6!
t6
w2
−
8!
3!5!
t5
w 3
+
8!
4!4!
t 4
w 4

40. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= t8
−
8!
1!7!
t7
w1
+
8!
2!6!
t6
w2
−
8!
3!5!
t5
w 3
+
8!
4!4!
t 4
w 4
−
8!
5!3!
t 3
w5

41. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= t8
−
8!
1!7!
t7
w1
+
8!
2!6!
t6
w2
−
8!
3!5!
t5
w 3
+
8!
4!4!
t 4
w 4
−
8!
5!3!
t 3
w5
+
8!
6!2!
t2
w6

42. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= t8
−
8!
1!7!
t7
w1
+
8!
2!6!
t6
w2
−
8!
3!5!
t5
w 3
+
8!
4!4!
t 4
w 4
−
8!
5!3!
t 3
w5
+
8!
6!2!
t2
w6
−
8!
7!1!
tw7

43. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= t8
−
8!
1!7!
t7
w1
+
8!
2!6!
t6
w2
−
8!
3!5!
t5
w 3
+
8!
4!4!
t 4
w 4
−
8!
5!3!
t 3
w5
+
8!
6!2!
t2
w6
−
8!
7!1!
tw7
+w8

44. Example 2
Expand (t −w)8
Binomial Theorem

45. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= t8

46. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= t8
−8t7
w

47. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= t8
−8t7
w +28t6
w2

48. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= t8
−8t7
w +28t6
w2
−56t5
w 3

49. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= t8
−8t7
w +28t6
w2
−56t5
w 3
+70t 4
w 4

50. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= t8
−8t7
w +28t6
w2
−56t5
w 3
+70t 4
w 4
−56t 3
w5

51. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= t8
−8t7
w +28t6
w2
−56t5
w 3
+70t 4
w 4
−56t 3
w5
+28t2
w6

52. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= t8
−8t7
w +28t6
w2
−56t5
w 3
+70t 4
w 4
−56t 3
w5
+28t2
w6
−8tw7

53. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= t8
−8t7
w +28t6
w2
−56t5
w 3
+70t 4
w 4
−56t 3
w5
+28t2
w6
−8tw7
+w8

54. Example 2
Expand (t −w)8
Binomial Theorem
(t −w)8
= t8
−8t7
w +28t6
w2
−56t5
w 3
+70t 4
w 4
−56t 3
w5
+28t2
w6
−8tw7
+w8
(t −w)8
= t8
− 8t7
w + 28t6
w2
− 56t5
w 3
+ 70t 4
w 4
− 56t 3
w5
+ 28t2
w6
− 8tw7
+w8

55. Example 2
Expand (t −w)8
Pascal’s Triangle

56. Example 2
Expand (t −w)8
Pascal’s Triangle
Find the ninth row of the triangle for the coefficients of
each term, make count down the exponents from the
first part of your binomial and count up the exponents
from the second term.

57. Example 2
Expand (t −w)8
Pascal’s Triangle
(t −w)8
= t8
− 8t7
w + 28t6
w2
− 56t5
w 3
+ 70t 4
w 4
− 56t 3
w5
+ 28t2
w6
− 8tw7
+w8
Find the ninth row of the triangle for the coefficients of
each term, make count down the exponents from the
first part of your binomial and count up the exponents
from the second term.

58. Example 3
Expand (a + 3b)4

59. Example 3
Expand (a + 3b)4
(a + 3b)4
= a4
+ 4a3
(3b)1
+ 6a2
(3b)2
+ 4a(3b)3
+ (3b)4

60. Example 3
Expand (a + 3b)4
(a + 3b)4
= a4
+ 4a3
(3b)1
+ 6a2
(3b)2
+ 4a(3b)3
+ (3b)4
(a + 3b)4
= a4
+ 4a3
i 3b + 6a2
i 9b2
+ 4a i 27b3
+ 81b4

61. Example 3
Expand (a + 3b)4
(a + 3b)4
= a4
+ 4a3
(3b)1
+ 6a2
(3b)2
+ 4a(3b)3
+ (3b)4
(a + 3b)4
= a4
+ 4a3
i 3b + 6a2
i 9b2
+ 4a i 27b3
+ 81b4
(a + 3b)4
= a4
+12a3
b + 54a2
b2
+108ab3
+ 81b4

62. Example 3
Find the third term of (3x − y )4

63. Example 3
Find the third term of (3x − y )4
a + b( )n
=
n!
k!(n − k )!
an−k
bk
k =0
n
∑

64. Example 3
Find the third term of (3x − y )4
a + b( )n
=
n!
k!(n − k )!
an−k
bk
k =0
n
∑
3x − y( )4
=
4!
k!(4 − k )!
(3x )4−k
(−y )k
k =0
4
∑

65. Example 3
Find the third term of (3x − y )4
a + b( )n
=
n!
k!(n − k )!
an−k
bk
k =0
n
∑
3x − y( )4
=
4!
k!(4 − k )!
(3x )4−k
(−y )k
k =0
4
∑
4!
2!(4 − 2)!
(3x )4−2
(−y )2

66. Example 3
Find the third term of (3x − y )4
a + b( )n
=
n!
k!(n − k )!
an−k
bk
k =0
n
∑
3x − y( )4
=
4!
k!(4 − k )!
(3x )4−k
(−y )k
k =0
4
∑
4!
2!(4 − 2)!
(3x )4−2
(−y )2
4!
2!2!
(3x )2
(−y )2

67. Example 3
Find the third term of (3x − y )4

68. Example 3
Find the third term of (3x − y )4
4!
2!2!
(3x )2
(−y )2

69. Example 3
Find the third term of (3x − y )4
4!
2!2!
(3x )2
(−y )2
6 i 9x 2
y 2

70. Example 3
Find the third term of (3x − y )4
4!
2!2!
(3x )2
(−y )2
6 i 9x 2
y 2
54x 2
y 2

71. Example 3
Find the third term of (3x − y )4

72. Example 3
Find the third term of (3x − y )4
1

73. Example 3
Find the third term of (3x − y )4
1
1 1

74. Example 3
Find the third term of (3x − y )4
1
1 1
1 12

75. Example 3
Find the third term of (3x − y )4
1
1 1
1 12
3 31 1

76. Example 3
Find the third term of (3x − y )4
1
1 1
1 12
3 31 1
1 4 4 16

77. Example 3
Find the third term of (3x − y )4
1
1 1
1 12
3 31 1
1 4 4 16

78. Example 3
Find the third term of (3x − y )4
1
1 1
1 12
3 31 1
1 4 4 16
6(3x )2
(−y )2

79. Example 3
Find the third term of (3x − y )4
1
1 1
1 12
3 31 1
1 4 4 16
6(3x )2
(−y )2
6 i 9x 2
y 2

80. Example 3
Find the third term of (3x − y )4
1
1 1
1 12
3 31 1
1 4 4 16
6(3x )2
(−y )2
6 i 9x 2
y 2
54x 2
y 2